On the energy of symmetric matrices and Coulson's integral formula

We dene the outer energy of a real symmetric matrix M for the eigenvalues λ₁, …, λ_n of M and their arithmetic mean λ(M). We discuss the properties of the outer energy in contrast to the inner energy defined as E_inn(M) = ∑ⁿ_{i = 1} |λ_i|. We prove that E_inn is the maximum among the energy functions e: S(n) → R and E_out among functions f(M - λ(M)1_n), where f is an energy function. We prove a variant of the Coulson integral formula for the outer energy.

DOI: https://doi.org/10.15446/recolma.v50n2.62209

Sobre la energía de matrices simétricas y la fórmula integral de Coulson

J. A. de la Peña¹, J. Rada²

¹ Centro de Investigación en Matemáticas, A.C., Guanajuato, México. jap@cimat.mx
² Universidad de Antioquia, Medellín, Colombia. pablo.rada@udea.edu.co

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Abstract

We define the outer energy of a real symmetric matrix M as for the eigenvalues λ₁, …, λ_n of M and their arithmetic mean . We discuss the properties of the outer energy in contrast to the inner energy defined as E_inn(M) = . We prove that E_inn is the maximum among the energy functions e: S(n) → R and E_out among functions f (M - 1_n, where f is an energy function. We prove a variant of the Coulson integral formula for the outer energy.

Keywords: Total π-electron energy, Energy of a symmetric matrix, Bounds for energy, Coulson's integral formula.

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Mathematics Subject Classification: 05C50.

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Resumen

Definimos la energía exterior de una matriz simétrica real M como donde λ₁, …, λ_n son los autovalores M y es su media aritmética. Discutimos las propiedades de la energía exterior en contraste con la energía interior definida como E_inn(M) = . Demostramos que E_inn es máxima entre todas las funciones de energía e: S(n) → R y E_out entre todas las funciones f (M - 1_n, donde f es una función de energía. Demostramos una variante de la fórmula integral de Coulson para la energía exterior.

Palabras claves: Energía π-electrón total, Energía de una matriz simétrica, Cotas para la energía, Fórmula integral de Coulson.

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Recibido: julio de 2016 Aceptado: septiembre de 2016